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→Chroma-diesis model of mos child scales: Included flattone temperament and how its "diesis" is larger than the chroma
Ganaram inukshuk (talk | contribs)
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→Chroma-diesis model of mos child scales: Finished up describing the chroma-diesis model
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This is a description of how to look at the child scales of a [[MOS scale|mos]] by looking at only the large and small steps of its parent mos. (It's also not well refined or proofread, hence it's a subpage of my userpage.) The motivation behind this comes from the notion of a [[chroma]] -- the interval that is defined as the difference between a mos's large and small steps -- and the [[diesis]], which can be defined as the difference between C# and Db in meantone temperaments.
This is a description of how to look at the child scales of a [[MOS scale|mos]] by looking at only the large and small steps of its parent mos. (It's also not well refined or proofread, hence it's a subpage of my userpage.) The motivation behind this comes from the notion of a [[chroma]] -- the interval that is defined as the difference between a mos's large and small steps -- and the [[diesis]], which can be defined as the difference between C# and Db in meantone temperaments.


This section describes the notion of a generalized diesis in both a temperament-related context and a temperament-agnostic context. (yay abuse of terminology)
This section describes the notion of a generalized diesis in both an [[regular temperament]] context and a [[TAMNAMS|temperament-agnostic]] context. I developed this model because I kept looking at child scales two generations after a parent scale, specifically 5L 2s and its children, and I needed a way to justify notating harmonic-7th chords (in meantone temperaments) as sharp-6 chords.


=== 7L 5s and 12L 7s (meantone temperament) ===
=== 7L 5s and 12L 7s (meantone temperament) ===
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Note that this model looks at child scales two generations beyond the parent scale. It's possible to generalize this to even smaller intervals (perhaps using a "triesis" defined as L - 3s and a general "polyesis" or "n-esis" defined as L - ns), but since the chroma and diesis are both familiar intervals (at least in a xen context), the named steps are limited to such, hence the name "chroma-diesis model".
Note that this model looks at child scales two generations beyond the parent scale. It's possible to generalize this to even smaller intervals (perhaps using a "triesis" defined as L - 3s and a general "polyesis" or "n-esis" defined as L - ns), but since the chroma and diesis are both familiar intervals (at least in a xen context), the named steps are limited to such, hence the name "chroma-diesis model".
Also note that the example of 31edo was chosen because its chroma-diesis ratio (its L:s ratio) is 2:1. Other edos can work as well, such as [[50edo]].


=== Including 7L 12s (flattone temperament) ===
=== Including 7L 12s (flattone temperament) ===
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In [[flattone]] temperament, it may be said that the diesis, as the difference between a small step and a chroma, is larger than the chroma; in comparison to meantone temperament, the diesis is smaller than the chroma. In a temperament-agnostic perspective, this is equivalent to describing a mos (7a 12b) without specifying which steps are the large or small steps, and specifying which is which will necessarily identify which of the two child mosses -- 7L 12s or 12L 7s -- is being described.
In [[flattone]] temperament, it may be said that the diesis, as the difference between a small step and a chroma, is larger than the chroma; in comparison to meantone temperament, the diesis is smaller than the chroma. In a temperament-agnostic perspective, this is equivalent to describing a mos (7a 12b) without specifying which steps are the large or small steps, and specifying which is which will necessarily identify which of the two child mosses -- 7L 12s or 12L 7s -- is being described.


=== Including 5L 7s, 5L 12s, and 12L 5s (pythagorean temperaments) ===
As with 31edo, 26edo was chosen because its L:s ratio is also 2:1.
The notion of chromas also apply to [[5L 7s]], the child mos of 5L 2s given a hard step ratio. Compared to soft step ratios (or meantone and flattone temperaments), hard step ratios produce chromas that are larger than the small step.
 
=== Including 5L 7s, 5L 12s, and 12L 5s (Pythagorean-based temperaments) ===
The notion of chromas also apply to [[5L 7s]], the child mos of 5L 2s given a hard step ratio. Compared to soft step ratios (or, when considering temperaments, meantone and flattone temperaments), hard step ratios produce chromas that are larger than the small step. Still, the notion of describing child scales as either chromas or dieses can still be done here. [[22edo]] and [[29edo]] are used as examples the same way 31edo and 26edo were used as examples: the child scales two generations after 5L 2s are of a step ratio of 2:1.
{| class="wikitable"
! colspan="22" |Step Visualization (using ionian for comparison)
!Mos
!Step Pattern
!TAMNAMS Name
!Temperament
|-
| colspan="4" |L
| colspan="4" |L
|s
| colspan="4" |L
| colspan="4" |L
| colspan="4" |L
|s
|5L 2s
|LLsLLLs
|diatonic
|[[superpyth]][7]
|-
| colspan="3" |c
|s
| colspan="3" |c
|s
|s
| colspan="3" |c
|s
| colspan="3" |c
|s
| colspan="3" |c
|s
|s
|5L 7s
|cs cs s cs cs cs s
|p-chromatic
|superpyth[12]
|-
| colspan="2" |d
|s
|s
| colspan="2" |d
|s
|s
|s
| colspan="2" |d
|s
|s
| colspan="2" |d
|s
|s
| colspan="2" |d
|s
|s
|s
|5L 12s
|dss dss s dss dss dss s
|unnamed
|superpyth[17]
|-
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
| colspan="4" |22edo
|}
{| class="wikitable"
! colspan="29" |Step Visualization (using ionian for comparison)
!Mos
!Step Pattern
!TAMNAMS Name
!Temperament
|-
| colspan="5" |L
| colspan="5" |L
| colspan="2" |s
| colspan="5" |L
| colspan="5" |L
| colspan="5" |L
| colspan="2" |s
|5L 2s
|LLsLLLs
|diatonic
|[[leapfrog]][7]
|-
| colspan="3" |c
| colspan="2" |s
| colspan="3" |c
| colspan="2" |s
| colspan="2" |s
| colspan="3" |c
| colspan="2" |s
| colspan="3" |c
| colspan="2" |s
| colspan="3" |c
| colspan="2" |s
| colspan="2" |s
|5L 7s
|cs cs s cs cs cs s
|p-chromatic
|leapfrog[12]
|-
|d
| colspan="2" |s
| colspan="2" |s
|d
| colspan="2" |s
| colspan="2" |s
| colspan="2" |s
|d
| colspan="2" |s
| colspan="2" |s
|d
| colspan="2" |s
| colspan="2" |s
|d
| colspan="2" |s
| colspan="2" |s
| colspan="2" |s
|12L 5s
|dss dss s dss dss dss s
|unnamed
|leapfrog[17]
|-
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
| colspan="4" |29edo
|}
 
=== Combined scale tree ===
The past few sections arbitrarily had 31edo, 26edo, 22edo, and 29edo selected for the sake of example. It should be noted that other edos could have worked for these example. Combining these into a scale tree removes the notion of being locked to a specific edo and reveals a more common pattern.
{| class="wikitable"
! colspan="2" |Parent scale
! colspan="3" |1st orderchild scales
! colspan="3" |2nd order child scales
|-
!Mos
!Step pattern
!Mos
!Step pattern
!Step condition
!Mos
!Step pattern
!Step condition
|-
| colspan="2" rowspan="3" |
| colspan="3" |
|'''5L 12s'''
|dss dss s dss dss dss s
|s < d
|-
|'''5L 7s'''
|cs cs s cs cs cs s
|s < c
|''17n-edo''
|sss sss s sss sss sss s
|s = d
|-
| colspan="3" |
|'''12L 5s'''
|dss dss s dss dss dss s
|s > d
|-
|'''5L 2s'''
|LLsLLLs
|''12n-edo''
|ss ss s ss ss ss s
|s = c
| colspan="3" |
|-
| colspan="2" rowspan="3" |
| colspan="3" |
|'''7L 12s'''
|ccd ccd cd ccd ccd ccd cd
|c < d
|-
|'''7L 5s'''
|cs cs s cs cs cs s
|s > c
|''19n-edo''
|ccc ccc cc ccc ccc ccc cc
|c = d
|-
| colspan="3" |
|'''12L 7s'''
|ccd ccd cd ccd ccd ccd cd
|c > d
|}
Combining all four tables into a scale tree reveals a few patterns:
 
* Sister scale pairs, such as 5L 7s and 7L 5s, are being described without a notion of which is the large or small step.
* Scales with a hard step ratio have 2nd-order child scales where the scales are described using dieses and small steps. Only the parent's small step persists as being one of the two step sizes.
* Scales with a soft step ratio have 2nd-order child scales where both the large and small step of the parent scale eventually break down into chromas and dieses. None of the parent's step sizes persist this far.
* Mos recursion becomes readily apparent, especially the chunking operation with the 2nd-generation children of a soft-step-ratio parent scale.
 
=== Generalized scale tree for nondiatonic (not 5L 2s) mosses ===
The chroma-diesis model also generalizes for nondiatonic mosses. Since these mosses are greatly underexplored (compared to diatonic), it's hard to generally know which scales have a similar status as diatonic (such as having a similar note count), and thus, it may be easier to describe such scales using only L's and s's and not chromas and dieses. Though a good place to start may be the sister mos of diatonic: antidiatonic, or [[2L 5s]]. The antiphrygian mode is assumed to be the "default" mode, the same way ionian is for diatonic.
{| class="wikitable"
! colspan="2" |Parent scale
! colspan="3" |1st orderchild scales
! colspan="3" |2nd order child scales
|-
!Mos
!Step pattern
!Mos
!Step pattern
!Step condition
!Mos
!Step pattern
!Step condition
|-
| colspan="2" rowspan="3" |
| colspan="3" |
|'''2L 9s'''
|dss s s s dss s s
|s < d
|-
|'''2L 7s'''
|cs s s s cs s s
|s < c
|''11n-edo''
|sss s s s sss s s
|s = d
|-
| colspan="3" |
|'''9L 2s'''
|dss s s s dss s s
|s > d
|-
|'''2L 5s'''
|LsssLss
|''9n-edo''
|ss s s s ss s s
|s = c
| colspan="3" |
|-
| colspan="2" rowspan="3" |
| colspan="3" |
|'''7L 9s'''
|ccdcd cd cd cd ccdcd cd cd
|c < d
|-
|'''7L 2s'''
|cs s s s cs s s
|s > c
|''16n-edo''
|ccccc cc cc cc ccccc cc cc
|c = d
|-
| colspan="3" |
|'''9L 7s'''
|ccdcd cd cd cd ccdcd cd cd
|c > d
|}
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